How to "flatten" the faces of a solid to level with its vertices?
A very unscientific way to do it would be to first create this shape:

Use the Array modifier in Object Offset mode, rotate the offset object 72° on the Z axis:

Rotate your mesh in Edit mode so that if fits perfectly with its instances:

Apply the modifier, duplicate, rotate the copy 180° on the Y axis, rotate 36° on the Z axis, merge the 2 meshes:

Bevel if you want more geometry: